2 (**************************************************************************)
5 (* ||A|| A project by Andrea Asperti *)
7 (* ||I|| Developers: *)
8 (* ||T|| The HELM team. *)
9 (* ||A|| http://helm.cs.unibo.it *)
11 (* \ / This file is distributed under the terms of the *)
12 (* v GNU General Public License Version 2 *)
14 (**************************************************************************)
16 include "ground_2/notation/relations/rafter_3.ma".
17 include "ground_2/relocation/rtmap_istot.ma".
19 (* RELOCATION MAP ***********************************************************)
21 coinductive after: relation3 rtmap rtmap rtmap ≝
22 | after_refl: ∀f1,f2,f,g1,g2,g.
23 after f1 f2 f → ↑f1 = g1 → ↑f2 = g2 → ↑f = g → after g1 g2 g
24 | after_push: ∀f1,f2,f,g1,g2,g.
25 after f1 f2 f → ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → after g1 g2 g
26 | after_next: ∀f1,f2,f,g1,g.
27 after f1 f2 f → ⫯f1 = g1 → ⫯f = g → after g1 f2 g
30 interpretation "relational composition (rtmap)"
31 'RAfter f1 f2 f = (after f1 f2 f).
33 definition H_after_inj: predicate rtmap ≝
35 ∀f,f21,f22. f1 ⊚ f21 ≘ f → f1 ⊚ f22 ≘ f → f21 ≗ f22.
37 (* Basic inversion lemmas ***************************************************)
39 lemma after_inv_ppx: ∀g1,g2,g. g1 ⊚ g2 ≘ g → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 →
40 ∃∃f. f1 ⊚ f2 ≘ f & ↑f = g.
41 #g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
42 [ #g2 #g #Hf #H1 #H2 #H #x1 #x2 #Hx1 #Hx2 destruct
43 >(injective_push … Hx1) >(injective_push … Hx2) -x2 -x1
44 /2 width=3 by ex2_intro/
45 | #g2 #g #_ #_ #H2 #_ #x1 #x2 #_ #Hx2 destruct
46 elim (discr_push_next … Hx2)
47 | #g #_ #H1 #_ #x1 #x2 #Hx1 #_ destruct
48 elim (discr_push_next … Hx1)
52 lemma after_inv_pnx: ∀g1,g2,g. g1 ⊚ g2 ≘ g → ∀f1,f2. ↑f1 = g1 → ⫯f2 = g2 →
53 ∃∃f. f1 ⊚ f2 ≘ f & ⫯f = g.
54 #g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
55 [ #g2 #g #_ #_ #H2 #_ #x1 #x2 #_ #Hx2 destruct
56 elim (discr_next_push … Hx2)
57 | #g2 #g #Hf #H1 #H2 #H3 #x1 #x2 #Hx1 #Hx2 destruct
58 >(injective_push … Hx1) >(injective_next … Hx2) -x2 -x1
59 /2 width=3 by ex2_intro/
60 | #g #_ #H1 #_ #x1 #x2 #Hx1 #_ destruct
61 elim (discr_push_next … Hx1)
65 lemma after_inv_nxx: ∀g1,f2,g. g1 ⊚ f2 ≘ g → ∀f1. ⫯f1 = g1 →
66 ∃∃f. f1 ⊚ f2 ≘ f & ⫯f = g.
67 #g1 #f2 #g * -g1 -f2 -g #f1 #f2 #f #g1
68 [ #g2 #g #_ #H1 #_ #_ #x1 #Hx1 destruct
69 elim (discr_next_push … Hx1)
70 | #g2 #g #_ #H1 #_ #_ #x1 #Hx1 destruct
71 elim (discr_next_push … Hx1)
72 | #g #Hf #H1 #H #x1 #Hx1 destruct
73 >(injective_next … Hx1) -x1
74 /2 width=3 by ex2_intro/
78 (* Advanced inversion lemmas ************************************************)
80 lemma after_inv_ppp: ∀g1,g2,g. g1 ⊚ g2 ≘ g →
81 ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ↑f = g → f1 ⊚ f2 ≘ f.
82 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_ppx … Hg … H1 H2) -g1 -g2
83 #x #Hf #Hx destruct <(injective_push … Hx) -f //
86 lemma after_inv_ppn: ∀g1,g2,g. g1 ⊚ g2 ≘ g →
87 ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ⫯f = g → ⊥.
88 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_ppx … Hg … H1 H2) -g1 -g2
89 #x #Hf #Hx destruct elim (discr_push_next … Hx)
92 lemma after_inv_pnn: ∀g1,g2,g. g1 ⊚ g2 ≘ g →
93 ∀f1,f2,f. ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → f1 ⊚ f2 ≘ f.
94 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_pnx … Hg … H1 H2) -g1 -g2
95 #x #Hf #Hx destruct <(injective_next … Hx) -f //
98 lemma after_inv_pnp: ∀g1,g2,g. g1 ⊚ g2 ≘ g →
99 ∀f1,f2,f. ↑f1 = g1 → ⫯f2 = g2 → ↑f = g → ⊥.
100 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_pnx … Hg … H1 H2) -g1 -g2
101 #x #Hf #Hx destruct elim (discr_next_push … Hx)
104 lemma after_inv_nxn: ∀g1,f2,g. g1 ⊚ f2 ≘ g →
105 ∀f1,f. ⫯f1 = g1 → ⫯f = g → f1 ⊚ f2 ≘ f.
106 #g1 #f2 #g #Hg #f1 #f #H1 #H elim (after_inv_nxx … Hg … H1) -g1
107 #x #Hf #Hx destruct <(injective_next … Hx) -f //
110 lemma after_inv_nxp: ∀g1,f2,g. g1 ⊚ f2 ≘ g →
111 ∀f1,f. ⫯f1 = g1 → ↑f = g → ⊥.
112 #g1 #f2 #g #Hg #f1 #f #H1 #H elim (after_inv_nxx … Hg … H1) -g1
113 #x #Hf #Hx destruct elim (discr_next_push … Hx)
116 lemma after_inv_pxp: ∀g1,g2,g. g1 ⊚ g2 ≘ g →
117 ∀f1,f. ↑f1 = g1 → ↑f = g →
118 ∃∃f2. f1 ⊚ f2 ≘ f & ↑f2 = g2.
119 #g1 * * [2: #m2] #g2 #g #Hg #f1 #f #H1 #H
120 [ elim (after_inv_pnp … Hg … H1 … H) -g1 -g -f1 -f //
121 | lapply (after_inv_ppp … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
125 lemma after_inv_pxn: ∀g1,g2,g. g1 ⊚ g2 ≘ g →
126 ∀f1,f. ↑f1 = g1 → ⫯f = g →
127 ∃∃f2. f1 ⊚ f2 ≘ f & ⫯f2 = g2.
128 #g1 * * [2: #m2] #g2 #g #Hg #f1 #f #H1 #H
129 [ lapply (after_inv_pnn … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
130 | elim (after_inv_ppn … Hg … H1 … H) -g1 -g -f1 -f //
134 lemma after_inv_xxp: ∀g1,g2,g. g1 ⊚ g2 ≘ g → ∀f. ↑f = g →
135 ∃∃f1,f2. f1 ⊚ f2 ≘ f & ↑f1 = g1 & ↑f2 = g2.
136 * * [2: #m1 ] #g1 #g2 #g #Hg #f #H
137 [ elim (after_inv_nxp … Hg … H) -g2 -g -f //
138 | elim (after_inv_pxp … Hg … H) -g /2 width=5 by ex3_2_intro/
142 lemma after_inv_xxn: ∀g1,g2,g. g1 ⊚ g2 ≘ g → ∀f. ⫯f = g →
143 (∃∃f1,f2. f1 ⊚ f2 ≘ f & ↑f1 = g1 & ⫯f2 = g2) ∨
144 ∃∃f1. f1 ⊚ g2 ≘ f & ⫯f1 = g1.
145 * * [2: #m1 ] #g1 #g2 #g #Hg #f #H
146 [ /4 width=5 by after_inv_nxn, or_intror, ex2_intro/
147 | elim (after_inv_pxn … Hg … H) -g
148 /3 width=5 by or_introl, ex3_2_intro/
152 lemma after_inv_pxx: ∀g1,g2,g. g1 ⊚ g2 ≘ g → ∀f1. ↑f1 = g1 →
153 (∃∃f2,f. f1 ⊚ f2 ≘ f & ↑f2 = g2 & ↑f = g) ∨
154 (∃∃f2,f. f1 ⊚ f2 ≘ f & ⫯f2 = g2 & ⫯f = g).
155 #g1 * * [2: #m2 ] #g2 #g #Hg #f1 #H
156 [ elim (after_inv_pnx … Hg … H) -g1
157 /3 width=5 by or_intror, ex3_2_intro/
158 | elim (after_inv_ppx … Hg … H) -g1
159 /3 width=5 by or_introl, ex3_2_intro/
163 (* Basic properties *********************************************************)
165 corec lemma after_eq_repl_back2: ∀f1,f. eq_repl_back (λf2. f2 ⊚ f1 ≘ f).
166 #f1 #f #f2 * -f2 -f1 -f
167 #f21 #f1 #f #g21 [1,2: #g1 ] #g #Hf #H21 [1,2: #H1 ] #H #g22 #H0
168 [ cases (eq_inv_px … H0 … H21) -g21 /3 width=7 by after_refl/
169 | cases (eq_inv_px … H0 … H21) -g21 /3 width=7 by after_push/
170 | cases (eq_inv_nx … H0 … H21) -g21 /3 width=5 by after_next/
174 lemma after_eq_repl_fwd2: ∀f1,f. eq_repl_fwd (λf2. f2 ⊚ f1 ≘ f).
175 #f1 #f @eq_repl_sym /2 width=3 by after_eq_repl_back2/
178 corec lemma after_eq_repl_back1: ∀f2,f. eq_repl_back (λf1. f2 ⊚ f1 ≘ f).
179 #f2 #f #f1 * -f2 -f1 -f
180 #f2 #f11 #f #g2 [1,2: #g11 ] #g #Hf #H2 [1,2: #H11 ] #H #g2 #H0
181 [ cases (eq_inv_px … H0 … H11) -g11 /3 width=7 by after_refl/
182 | cases (eq_inv_nx … H0 … H11) -g11 /3 width=7 by after_push/
183 | @(after_next … H2 H) /2 width=5 by/
187 lemma after_eq_repl_fwd1: ∀f2,f. eq_repl_fwd (λf1. f2 ⊚ f1 ≘ f).
188 #f2 #f @eq_repl_sym /2 width=3 by after_eq_repl_back1/
191 corec lemma after_eq_repl_back0: ∀f1,f2. eq_repl_back (λf. f2 ⊚ f1 ≘ f).
192 #f2 #f1 #f * -f2 -f1 -f
193 #f2 #f1 #f01 #g2 [1,2: #g1 ] #g01 #Hf01 #H2 [1,2: #H1 ] #H01 #g02 #H0
194 [ cases (eq_inv_px … H0 … H01) -g01 /3 width=7 by after_refl/
195 | cases (eq_inv_nx … H0 … H01) -g01 /3 width=7 by after_push/
196 | cases (eq_inv_nx … H0 … H01) -g01 /3 width=5 by after_next/
200 lemma after_eq_repl_fwd0: ∀f2,f1. eq_repl_fwd (λf. f2 ⊚ f1 ≘ f).
201 #f2 #f1 @eq_repl_sym /2 width=3 by after_eq_repl_back0/
204 (* Main properties **********************************************************)
206 corec theorem after_trans1: ∀f0,f3,f4. f0 ⊚ f3 ≘ f4 →
207 ∀f1,f2. f1 ⊚ f2 ≘ f0 →
208 ∀f. f2 ⊚ f3 ≘ f → f1 ⊚ f ≘ f4.
209 #f0 #f3 #f4 * -f0 -f3 -f4 #f0 #f3 #f4 #g0 [1,2: #g3 ] #g4
210 [ #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
211 cases (after_inv_xxp … Hg0 … H0) -g0
213 cases (after_inv_ppx … Hg … H2 H3) -g2 -g3
214 #f #Hf #H /3 width=7 by after_refl/
215 | #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
216 cases (after_inv_xxp … Hg0 … H0) -g0
218 cases (after_inv_pnx … Hg … H2 H3) -g2 -g3
219 #f #Hf #H /3 width=7 by after_push/
220 | #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
221 cases (after_inv_xxn … Hg0 … H0) -g0 *
222 [ #f1 #f2 #Hf0 #H1 #H2
223 cases (after_inv_nxx … Hg … H2) -g2
224 #f #Hf #H /3 width=7 by after_push/
225 | #f1 #Hf0 #H1 /3 width=6 by after_next/
230 corec theorem after_trans2: ∀f1,f0,f4. f1 ⊚ f0 ≘ f4 →
231 ∀f2, f3. f2 ⊚ f3 ≘ f0 →
232 ∀f. f1 ⊚ f2 ≘ f → f ⊚ f3 ≘ f4.
233 #f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #g1 [1,2: #g0 ] #g4
234 [ #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
235 cases (after_inv_xxp … Hg0 … H0) -g0
237 cases (after_inv_ppx … Hg … H1 H2) -g1 -g2
238 #f #Hf #H /3 width=7 by after_refl/
239 | #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
240 cases (after_inv_xxn … Hg0 … H0) -g0 *
241 [ #f2 #f3 #Hf0 #H2 #H3
242 cases (after_inv_ppx … Hg … H1 H2) -g1 -g2
243 #f #Hf #H /3 width=7 by after_push/
245 cases (after_inv_pnx … Hg … H1 H2) -g1 -g2
246 #f #Hf #H /3 width=6 by after_next/
248 | #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
249 cases (after_inv_nxx … Hg … H1) -g1
250 #f #Hg #H /3 width=6 by after_next/
254 (* Main inversion lemmas ****************************************************)
256 corec theorem after_mono: ∀f1,f2,x,y. f1 ⊚ f2 ≘ x → f1 ⊚ f2 ≘ y → x ≗ y.
257 #f1 #f2 #x #y * -f1 -f2 -x
258 #f1 #f2 #x #g1 [1,2: #g2 ] #g #Hx #H1 [1,2: #H2 ] #H0x #Hy
259 [ cases (after_inv_ppx … Hy … H1 H2) -g1 -g2 /3 width=8 by eq_push/
260 | cases (after_inv_pnx … Hy … H1 H2) -g1 -g2 /3 width=8 by eq_next/
261 | cases (after_inv_nxx … Hy … H1) -g1 /3 width=8 by eq_next/
265 lemma after_mono_eq: ∀f1,f2,f. f1 ⊚ f2 ≘ f → ∀g1,g2,g. g1 ⊚ g2 ≘ g →
266 f1 ≗ g1 → f2 ≗ g2 → f ≗ g.
267 /4 width=4 by after_mono, after_eq_repl_back1, after_eq_repl_back2/ qed-.
269 (* Properties on tls ********************************************************)
271 lemma after_tls: ∀n,f1,f2,f. @⦃0, f1⦄ ≘ n →
272 f1 ⊚ f2 ≘ f → ⫱*[n]f1 ⊚ f2 ≘ ⫱*[n]f.
274 #n #IH #f1 #f2 #f #Hf1 #Hf
275 cases (at_inv_pxn … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1
276 cases (after_inv_nxx … Hf … H1) -Hf #g #Hg #H0 destruct
277 <tls_xn <tls_xn /2 width=1 by/
280 (* Properties on isid *******************************************************)
282 corec lemma after_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ⊚ f2 ≘ f2.
283 #f1 * -f1 #f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) * #g2 #H2
284 /3 width=7 by after_push, after_refl/
287 corec lemma after_isid_dx: ∀f2. 𝐈⦃f2⦄ → ∀f1. f1 ⊚ f2 ≘ f1.
288 #f2 * -f2 #f2 #g2 #Hf2 #H2 #f1 cases (pn_split f1) * #g1 #H1
289 [ /3 width=7 by after_refl/
290 | @(after_next … H1 H1) /3 width=3 by isid_push/
294 (* Inversion lemmas on isid *************************************************)
296 lemma after_isid_inv_sn: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈⦃f1⦄ → f2 ≗ f.
297 /3 width=6 by after_isid_sn, after_mono/ qed-.
299 lemma after_isid_inv_dx: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈⦃f2⦄ → f1 ≗ f.
300 /3 width=6 by after_isid_dx, after_mono/ qed-.
302 corec lemma after_fwd_isid1: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄.
303 #f1 #f2 #f * -f1 -f2 -f
304 #f1 #f2 #f #g1 [1,2: #g2 ] #g #Hf #H1 [1,2: #H2 ] #H0 #H
305 [ /4 width=6 by isid_inv_push, isid_push/ ]
306 cases (isid_inv_next … H … H0)
309 corec lemma after_fwd_isid2: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈⦃f⦄ → 𝐈⦃f2⦄.
310 #f1 #f2 #f * -f1 -f2 -f
311 #f1 #f2 #f #g1 [1,2: #g2 ] #g #Hf #H1 [1,2: #H2 ] #H0 #H
312 [ /4 width=6 by isid_inv_push, isid_push/ ]
313 cases (isid_inv_next … H … H0)
316 lemma after_inv_isid3: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄ ∧ 𝐈⦃f2⦄.
317 /3 width=4 by after_fwd_isid2, after_fwd_isid1, conj/ qed-.
319 (* Properties on isuni ******************************************************)
321 lemma after_isid_isuni: ∀f1,f2. 𝐈⦃f2⦄ → 𝐔⦃f1⦄ → f1 ⊚ ⫯f2 ≘ ⫯f1.
322 #f1 #f2 #Hf2 #H elim H -H
323 /5 width=7 by after_isid_dx, after_eq_repl_back2, after_next, after_push, eq_push_inv_isid/
326 lemma after_uni_next2: ∀f2. 𝐔⦃f2⦄ → ∀f1,f. ⫯f2 ⊚ f1 ≘ f → f2 ⊚ ⫯f1 ≘ f.
328 [ #f2 #Hf2 #f1 #f #Hf
329 elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
330 /4 width=7 by after_isid_inv_sn, after_isid_sn, after_eq_repl_back0, eq_next/
331 | #f2 #_ #g2 #H2 #IH #f1 #f #Hf
332 elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
333 /3 width=5 by after_next/
337 (* Properties on uni ********************************************************)
339 lemma after_uni: ∀n1,n2. 𝐔❴n1❵ ⊚ 𝐔❴n2❵ ≘ 𝐔❴n1+n2❵.
340 @nat_elim2 [3: #n #m <plus_n_Sm ] (**) (* full auto fails *)
341 /4 width=5 by after_uni_next2, after_isid_dx, after_isid_sn, after_next/
344 (* Forward lemmas on at *****************************************************)
346 lemma after_at_fwd: ∀i,i1,f. @⦃i1, f⦄ ≘ i → ∀f2,f1. f2 ⊚ f1 ≘ f →
347 ∃∃i2. @⦃i1, f1⦄ ≘ i2 & @⦃i2, f2⦄ ≘ i.
348 #i elim i -i [2: #i #IH ] #i1 #f #Hf #f2 #f1 #Hf21
349 [ elim (at_inv_xxn … Hf) -Hf [1,3:* |*: // ]
350 [1: #g #j1 #Hg #H0 #H |2,4: #g #Hg #H ]
351 | elim (at_inv_xxp … Hf) -Hf //
354 [2: elim (after_inv_xxn … Hf21 … H) -f *
355 [ #g2 #g1 #Hg21 #H2 #H1 | #g2 #Hg21 #H2 ]
356 |*: elim (after_inv_xxp … Hf21 … H) -f
357 #g2 #g1 #Hg21 #H2 #H1
359 [4: -Hg21 |*: elim (IH … Hg … Hg21) -g -IH ]
360 /3 width=9 by at_refl, at_push, at_next, ex2_intro/
363 lemma after_fwd_at: ∀i,i2,i1,f1,f2. @⦃i1, f1⦄ ≘ i2 → @⦃i2, f2⦄ ≘ i →
364 ∀f. f2 ⊚ f1 ≘ f → @⦃i1, f⦄ ≘ i.
365 #i elim i -i [2: #i #IH ] #i2 #i1 #f1 #f2 #Hf1 #Hf2 #f #Hf
366 [ elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
367 #g2 [ #j2 ] #Hg2 [ #H22 ] #H20
368 [ elim (at_inv_xxn … Hf1 … H22) -i2 *
369 #g1 [ #j1 ] #Hg1 [ #H11 ] #H10
370 [ elim (after_inv_ppx … Hf … H20 H10) -f1 -f2 /3 width=7 by at_push/
371 | elim (after_inv_pnx … Hf … H20 H10) -f1 -f2 /3 width=6 by at_next/
373 | elim (after_inv_nxx … Hf … H20) -f2 /3 width=7 by at_next/
375 | elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H22 #H20
376 elim (at_inv_xxp … Hf1 … H22) -i2 #g1 #H11 #H10
377 elim (after_inv_ppx … Hf … H20 H10) -f1 -f2 /2 width=2 by at_refl/
381 lemma after_fwd_at2: ∀f,i1,i. @⦃i1, f⦄ ≘ i → ∀f1,i2. @⦃i1, f1⦄ ≘ i2 →
382 ∀f2. f2 ⊚ f1 ≘ f → @⦃i2, f2⦄ ≘ i.
383 #f #i1 #i #Hf #f1 #i2 #Hf1 #f2 #H elim (after_at_fwd … Hf … H) -f
384 #j1 #H #Hf2 <(at_mono … Hf1 … H) -i1 -i2 //
387 lemma after_fwd_at1: ∀i,i2,i1,f,f2. @⦃i1, f⦄ ≘ i → @⦃i2, f2⦄ ≘ i →
388 ∀f1. f2 ⊚ f1 ≘ f → @⦃i1, f1⦄ ≘ i2.
389 #i elim i -i [2: #i #IH ] #i2 #i1 #f #f2 #Hf #Hf2 #f1 #Hf1
390 [ elim (at_inv_xxn … Hf) -Hf [1,3: * |*: // ]
391 #g [ #j1 ] #Hg [ #H01 ] #H00
392 elim (at_inv_xxn … Hf2) -Hf2 [1,3,5,7: * |*: // ]
393 #g2 [1,3: #j2 ] #Hg2 [1,2: #H22 ] #H20
394 [ elim (after_inv_pxp … Hf1 … H20 H00) -f2 -f /3 width=7 by at_push/
395 | elim (after_inv_pxn … Hf1 … H20 H00) -f2 -f /3 width=5 by at_next/
396 | elim (after_inv_nxp … Hf1 … H20 H00)
397 | /4 width=9 by after_inv_nxn, at_next/
399 | elim (at_inv_xxp … Hf) -Hf // #g #H01 #H00
400 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H21 #H20
401 elim (after_inv_pxp … Hf1 … H20 H00) -f2 -f /3 width=2 by at_refl/
405 (* Properties with at *******************************************************)
407 lemma after_uni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
408 ∀f. f2 ⊚ 𝐔❴i1❵ ≘ f → 𝐔❴i2❵ ⊚ ⫱*[i2] f2 ≘ f.
410 [ #i1 #f2 #Hf2 #f #Hf
411 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
412 lapply (after_isid_inv_dx … Hf ?) -Hf
413 /3 width=3 by after_isid_sn, after_eq_repl_back0/
414 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
415 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
416 [ #g2 #j1 #Hg2 #H1 #H2 destruct
417 elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
418 <tls_xn /3 width=5 by after_next/
419 | #g2 #Hg2 #H2 destruct
420 elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
421 <tls_xn /3 width=5 by after_next/
426 lemma after_uni_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
427 ∀f. 𝐔❴i2❵ ⊚ ⫱*[i2] f2 ≘ f → f2 ⊚ 𝐔❴i1❵ ≘ f.
429 [ #i1 #f2 #Hf2 #f #Hf
430 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
431 lapply (after_isid_inv_sn … Hf ?) -Hf
432 /3 width=3 by after_isid_dx, after_eq_repl_back0/
433 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
434 elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
435 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
436 [ #g2 #j1 #Hg2 #H1 #H2 destruct /3 width=7 by after_push/
437 | #g2 #Hg2 #H2 destruct /3 width=5 by after_next/
442 lemma after_uni_succ_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
443 ∀f. f2 ⊚ 𝐔❴⫯i1❵ ≘ f → 𝐔❴⫯i2❵ ⊚ ⫱*[⫯i2] f2 ≘ f.
445 [ #i1 #f2 #Hf2 #f #Hf
446 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
447 elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H
448 lapply (after_isid_inv_dx … Hg ?) -Hg
449 /4 width=5 by after_isid_sn, after_eq_repl_back0, after_next/
450 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
451 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
452 [ #g2 #j1 #Hg2 #H1 #H2 destruct
453 elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
454 <tls_xn /3 width=5 by after_next/
455 | #g2 #Hg2 #H2 destruct
456 elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
457 <tls_xn /3 width=5 by after_next/
462 lemma after_uni_succ_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
463 ∀f. 𝐔❴⫯i2❵ ⊚ ⫱*[⫯i2] f2 ≘ f → f2 ⊚ 𝐔❴⫯i1❵ ≘ f.
465 [ #i1 #f2 #Hf2 #f #Hf
466 elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
467 elim (after_inv_nxx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
468 lapply (after_isid_inv_sn … Hg ?) -Hg
469 /4 width=7 by after_isid_dx, after_eq_repl_back0, after_push/
470 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
471 elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
472 elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
473 [ #g2 #j1 #Hg2 #H1 #H2 destruct <tls_xn in Hg; /3 width=7 by after_push/
474 | #g2 #Hg2 #H2 destruct <tls_xn in Hg; /3 width=5 by after_next/
479 lemma after_uni_one_dx: ∀f2,f. ↑f2 ⊚ 𝐔❴⫯O❵ ≘ f → 𝐔❴⫯O❵ ⊚ f2 ≘ f.
480 #f2 #f #H @(after_uni_succ_dx … (↑f2)) /2 width=3 by at_refl/
483 lemma after_uni_one_sn: ∀f1,f. 𝐔❴⫯O❵ ⊚ f1 ≘ f → ↑f1 ⊚ 𝐔❴⫯O❵ ≘ f.
484 /3 width=3 by after_uni_succ_sn, at_refl/ qed-.
486 (* Forward lemmas on istot **************************************************)
488 lemma after_istot_fwd: ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓⦃f2⦄ → 𝐓⦃f1⦄ → 𝐓⦃f⦄.
489 #f2 #f1 #f #Hf #Hf2 #Hf1 #i1 elim (Hf1 i1) -Hf1
490 #i2 #Hf1 elim (Hf2 i2) -Hf2
491 /3 width=7 by after_fwd_at, ex_intro/
494 lemma after_fwd_istot_dx: ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓⦃f⦄ → 𝐓⦃f1⦄.
495 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
496 #i2 #Hf elim (after_at_fwd … Hf … H) -f /2 width=2 by ex_intro/
499 lemma after_fwd_istot_sn: ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓⦃f⦄ → 𝐓⦃f2⦄.
500 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
501 #i #Hf elim (after_at_fwd … Hf … H) -f
502 #i2 #Hf1 #Hf2 lapply (at_increasing … Hf1) -f1
503 #Hi12 elim (at_le_ex … Hf2 … Hi12) -i2 /2 width=2 by ex_intro/
506 lemma after_inv_istot: ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓⦃f⦄ → 𝐓⦃f2⦄ ∧ 𝐓⦃f1⦄.
507 /3 width=4 by after_fwd_istot_sn, after_fwd_istot_dx, conj/ qed-.
509 lemma after_at1_fwd: ∀f1,i1,i2. @⦃i1, f1⦄ ≘ i2 → ∀f2. 𝐓⦃f2⦄ → ∀f. f2 ⊚ f1 ≘ f →
510 ∃∃i. @⦃i2, f2⦄ ≘ i & @⦃i1, f⦄ ≘ i.
511 #f1 #i1 #i2 #Hf1 #f2 #Hf2 #f #Hf elim (Hf2 i2) -Hf2
512 /3 width=8 by after_fwd_at, ex2_intro/
515 lemma after_fwd_isid_sn: ∀f2,f1,f. 𝐓⦃f⦄ → f2 ⊚ f1 ≘ f → f1 ≗ f → 𝐈⦃f2⦄.
516 #f2 #f1 #f #H #Hf elim (after_inv_istot … Hf H) -H
517 #Hf2 #Hf1 #H @isid_at_total // -Hf2
518 #i2 #i #Hf2 elim (Hf1 i2) -Hf1
519 #i0 #Hf1 lapply (at_increasing … Hf1)
520 #Hi20 lapply (after_fwd_at2 … i0 … Hf1 … Hf) -Hf
521 /3 width=7 by at_eq_repl_back, at_mono, at_id_le/
524 lemma after_fwd_isid_dx: ∀f2,f1,f. 𝐓⦃f⦄ → f2 ⊚ f1 ≘ f → f2 ≗ f → 𝐈⦃f1⦄.
525 #f2 #f1 #f #H #Hf elim (after_inv_istot … Hf H) -H
526 #Hf2 #Hf1 #H2 @isid_at_total // -Hf1
527 #i1 #i2 #Hi12 elim (after_at1_fwd … Hi12 … Hf) -f1
528 /3 width=8 by at_inj, at_eq_repl_back/
531 corec fact after_inj_O_aux: ∀f1. @⦃0, f1⦄ ≘ 0 → H_after_inj f1.
532 #f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
533 cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
534 lapply (istot_inv_push … H2f1 … H1) -H2f1 #H2g1
535 cases (H2g1 0) #n #Hn
536 cases (after_inv_pxx … H1f … H1) -H1f * #g21 #g #H1g #H21 #H
537 [ cases (after_inv_pxp … H2f … H1 H) -f1 -f #g22 #H2g #H22
538 @(eq_push … H21 H22) -f21 -f22
539 | cases (after_inv_pxn … H2f … H1 H) -f1 -f #g22 #H2g #H22
540 @(eq_next … H21 H22) -f21 -f22
542 @(after_inj_O_aux (⫱*[n]g1) … (⫱*[n]g)) -after_inj_O_aux
543 /2 width=1 by after_tls, istot_tls, at_pxx_tls/
546 fact after_inj_aux: (∀f1. @⦃0, f1⦄ ≘ 0 → H_after_inj f1) →
547 ∀i2,f1. @⦃0, f1⦄ ≘ i2 → H_after_inj f1.
548 #H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
549 #i2 #IH #f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
550 elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #H1g1 #H1
551 elim (after_inv_nxx … H1f … H1) -H1f #g #H1g #H
552 lapply (after_inv_nxn … H2f … H1 H) -f #H2g
553 /3 width=6 by istot_inv_next/
556 theorem after_inj: ∀f1. H_after_inj f1.
557 #f1 #H cases (H 0) /3 width=7 by after_inj_aux, after_inj_O_aux/